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Sum of all compositions [c_1, c_2, ..., c_q] of n encoded as Product_{i=1..q} prime(i)^(c_i).
3

%I #21 Apr 27 2021 10:56:45

%S 1,2,10,68,640,8372,147820,3321908,90184300,2857153892,104146026820,

%T 4363900557128,209763325978480,11462371025215112,702793156696129600,

%U 47649412958404240688,3521160558576929028400,280427910532671712997732,23932837097476310995036900

%N Sum of all compositions [c_1, c_2, ..., c_q] of n encoded as Product_{i=1..q} prime(i)^(c_i).

%H Alois P. Heinz, <a href="/A325054/b325054.txt">Table of n, a(n) for n = 0..335</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/G%C3%B6del_numbering#G%C3%B6del&#39;s_encoding">Gödel's encoding</a>

%e The compositions of 3 and their encodings are [1,1,1]-> 30, [1,2]-> 18, [2,1]-> 12, [3]-> 8. The sum gives a(3) = 68.

%p b:= proc(n, j) option remember; `if`(n=0, 1,

%p add(ithprime(j)^i*b(n-i, j+1), i=1..n))

%p end:

%p a:= n-> b(n, 1):

%p seq(a(n), n=0..20); # _Alois P. Heinz_, Sep 04 2019

%t b[n_, j_] := b[n, j] = If[n==0, 1, Sum[Prime[j]^i*b[n-i, j+1], {i, 1, n}]];

%t a[n_] := b[n, 1];

%t a /@ Range[0, 20] (* _Jean-François Alcover_, Apr 23 2021, after _Alois P. Heinz_ *)

%Y Row sums of A324939.

%Y Cf. A000040, A074140.

%K nonn

%O 0,2

%A _Alois P. Heinz_, Sep 04 2019